Yes, it is referred to as Nonenal.

"Old person smell" as it is popularly referred to is actually the production of an odor called Nonenal. Nonenal production increases after the age of 40 and is a natural part of human aging. Old person smell is not the result of poor hygiene or products that only old people use, but rather overproduction of Nonenal. It can be treated with certain foods and soaps, but stopping the overproduction of Nonenal after a certain age is not possible yet. Old person smell is not a myth, though.

Regardless of age, we all smell unique

The body emits orders depending on age and health, environment and diet. To say that a particular scent is completely identifiable with a subset of population based on only one criteria is not representative of the group at large. Many elderly, forced into a narrow band of living conditions with diets provided by the same group of cost-driven vendors are likely to develop a similarity it odor that would not apply to those living on their own terms.

Older people smell no different than others.

There is nothing that gives older people a different smell than younger people. Some things that people may notice, though, is that older people have lost their sense of smell and some aren't able to perform hygiene as well as before. This can sometimes contribute to an older person smelling differently, but it doesn't pertain to all older people, and isn't a distinct smell that they all share.

People that bathed properly smell bad

I don't think it's fair to stereotype older individuals. Anyone can smell bad when they don't bathe, so it has nothing to do with age. A younger person who skips a few showers would smell just as bad as someone up in age. It is way too easy to vilify people based on whatever criteria we choose based on various prejudices. It is a habit we have to fight against.